Renewable Energy 45 (2012) 31e40
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Renewable Energy
journal homepage: www.elsevier.com/locate/renene
On the annual wave energy absorption by two-body heaving WECs with latching
control
J.C.C. Henriques
a,
*
, M.F.P. Lopes
b
, R.P.F. Gomes
a
, L.M.C. Gato
a
, A.F.O. Falcão
a
a
b
IDMEC, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Wave Energy Centre, Av. Manuel da Maia, 1049-001 Lisboa, Portugal
a r t i c l e i n f o
Article history:
Received 13 October 2011
Accepted 31 January 2012
Available online 23 March 2012
Keywords:
Wave energy
Time domain
Latching control
Threshold unlatching time
Two-body system
Cummins equations
a b s t r a c t
Although the latching control strategy has been recognized as an important mean of increasing the
efficiency of one-body point-absorbing wave energy converters (WECs), its effectiveness in two-body
floating
point-absorbers has been questioned in some studies. The current work investigates the
increase in annual absorbed energy achieved with a simple threshold unlatching control strategy when
applied to a generic two-body heaving WEC. The WEC performance is evaluated for a set of sea-states
characteristic of the wave climate off the Portuguese west coast.
To achieve this computationally intensive task, a new high-order numerical method for the solution of
the Cummins equations is presented and used. This approach is based on a polynomial representation of
the solution, whose coefficients are computed using a continuous least-squares approximation. The code
has been parallelized and computations were performed at the IST cluster.
Ó
2012 Elsevier Ltd. All rights reserved.
1. Introduction
The energy extraction by offshore WECs is, in general, obtained
through the relative motion of two or more parts of the converter.
The moving parts may have relative motion in translation modes, as
in the Wavebob
[1]
and the IPS Buoy
[2],
or the relative rotation
between parts of the converter, as the case of SEAREV
[3].
Other
particular cases, as devices connected directly to the seabed and
bottom mounted oscillating water columns can be seen as partic-
ular cases of the
first
one, where the second body (the earth) has
infinite mass.
The two-body problem in heave is very relevant in the scope of
offshore wave energy conversion. This problem has been studied
by a number of authors both theoretically
[4,5]
and numerically
[6e9].
The latching control technique,
first
proposed by
[10]
for one-
body systems, consists in locking the body motion when its
velocity vanishes and releasing it such that ideally the velocity
becomes in phase with the excitation force during the unlatched
part of the cycle. Under certain conditions, a significant increase in
*
Corresponding author. Tel.:
þ351
218417296.
E-mail addresses:
joaochenriques@ist.utl.pt
(J.C.C. Henriques),
mlopes@wave-
energy-centre.org
(M.F.P. Lopes),
ruigomes@ist.utl.pt
(R.P.F. Gomes),
luis.gato@
ist.utl.pt
(L.M.C. Gato),
antonio.falcao@ist.utl.pt
(A.F.O. Falcão).
0960-1481/$
e
see front matter
Ó
2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.renene.2012.01.102
the energy extraction can be achieved, when compared with the
non-controlled case. This strategy can be adapted for the relative
motion of two-body systems, using the relative velocity as
reference.
The possibility offered by latching control of achieving larger
efficiencies over a broad range of incident frequencies can signifi-
cantly improve the device economics. Moreover, the efficiency gain
is larger at frequencies smaller than the device resonance
frequency, which may result in a substantially smaller device for
the same wave energy absorption capability
[11].
The practical
application of the latching strategy raises several problems, mainly
related to the need of fast response of mechanical and/or hydraulic
components. The study of this strategy for one-body systems has
been quite extensive including successful experimental investiga-
tions
[12e15].
The utilization of latching control in two-body heaving systems
has been justifiably questioned (see e.g.
[16])
because the two
bodies keep moving as one-body after latching, which makes the
system less efficient as the dynamics becomes different from what
was idealized in one-body latching control. However, the effect of
this is dependent on the mass ratio of the two bodies. In the
limiting case, when the mass of one of the bodies increases to
infinity, the gain from latching control increases since the system
becomes equivalent to a
floater
reacting against the seabed
[4].
This
paper examines the influence of the relative mass of the two bodies
on the latching control efficiency for a given annual wave climate.
32
J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40
2. Description of the problem
The present paper is focused on the performance of a generic
two-body heaving WEC with latching control. The device consists
of a
floating
hemisphere (body 1) rigidly connected to a deeply
submerged body (body 2), see
Fig. 1.
The two bodies are only
allowed to move in heave and they are connected through a power
take-off system (PTO) modelled as a linear damper. It is assumed
that the distance from the submerged body to the free surface is
large enough for the excitation and radiation forces to be negligible.
Furthermore, the hydrodynamic interaction between the
floater
and the submerged body is neglected. The latching strategy used
here consists in locking the bodies when their relative velocity
vanishes and releasing them at a specified time (threshold
unlatching) after the sign of the
floater
excitation force (assumed
known) multiplied by its vertical coordinate becomes negative. This
strategy is based on the results presented in
[2]
and
[15].
The
parameters to be studied are the ratio between the
floater
mass and
the submerged body inertial mass (body mass plus the added mass
of surrounding
fluid),
the PTO damping coefficient and the
threshold unlatching time. In each investigation, the ratio between
the
floater
mass and the mass plus added mass of the submerged
body is kept
fixed,
while the other parameters are allowed to vary
in order to
find
how to improve the annual energy absorption of the
device. In this work, it is investigated how to optimize two-
heaving-body point absorbers with latching control. A criterion is
established for the ratio between the
floater
mass and the mass
plus added mass of the submerged body if this type of control is to
be effective.
The WEC response is computed for a set of 14 sea-states char-
acterizing an Atlantic location off the western coast of Portugal.
Time series of 120 min were synthesized for each sea state,
considering a Pierson-Moskowitz spectrum. The energy absorbed
in each sea state is then weighted according to the sea state
probability of occurrence in order to estimate the annual averaged
power output.
The optimization of one-body WEC with latching control
regarding a specific wave climate has been presented in
[17].
In the
current work, contour plots for the dimensionless annual averaged
power output are shown for varying threshold unlatching time and
the PTO damping. Although it requires more computational time,
more information about the system behaviour can be retrieved
from these contour plots than from just the results from the
optimum operating point calculation.
Since latching implies a non-linear process, the equations of
motion must be solved in the time domain. In this paper, a new high-
order numerical method for the solution of the Cummins equations
[18]
is presented. This approach is based on a continuous polynomial
representation of the solution whose coefficients are computed
using a continuous least-squares approximation. A six-order poly-
nomial was used for the computation of the bodies’ positions. The
convolution integral is calculated using the continuous polynomial
approximation of the velocity as computed by the least-squares
method. This constitutes an important advantage of the proposed
method in comparison to the usual high-order Runge-Kutta
methods, where the convolution integral is computed with the
trapezoidal rule which limits the overall accuracy of the solution to
second-order. In the present approach, the convolution integral is
computed using a Gauss-Legendre quadrature rule
[19]
which
results in a product of a matrix by a vector of instantaneous
velocities.
The code has been successfully parallelized using the OpenMPI
library
[20]
to largely reduce the computational time. For the
computation of the annual energy absorption, each spectrum is
assigned to one processor. Optimal speed-up was achieved since
the amount of information transferred between processors is very
small. The computations were performed at the IST cluster.
3. Two-body WEC modelling
3.1. Governing equations
Consider the two-body heaving WEC represented in
Fig. 1.
Let
y
and
x
be the coordinates for positions of the heaving
floater
and
submerged body, with
y
¼
x
¼
0 at equilibrium position and both
increasing upwards. The heaving
floater
is subjected to a hydrody-
namic force,
F
y,
and a force transmitted through the PTO to the
submerged body,
F
PTO
. During latching, a braking force
F
brake
is
applied to keep both bodies
fixed
to each other (zero relative
velocity). The equations of motion for the two-body system are
À
Á
À
Á
€
¼
F
y
t; y; y; y
þ
F
PTO
y; x
þ
F
brake
t; y; x
;
_ €
_ _
_ _
m
1
y
À
Á
À
Á
_ _
_ _
x
ðm
2
þ
A
2
Þ
€
¼ ÀF
PTO
y; x
À
F
brake
t; y; x
;
(1)
where
m
i
is the mass of body
i
(i
¼
1,2). Since body 2 is assumed to
be deeply submerged, it does not radiate waves and its added mass
A
2
is independent of the frequency of motion.
The hydrodynamic force
F
y
is the vertical component of the force
due to water pressure on the
floater
wetted surface (F
y
¼
0 for
a motionless
floater
at
y
¼
0 in calm water). The force
F
y
depends on
known quantities. According to the linear wave theory, if the
amplitudes of the wave and of the
floater
motion are small, then the
hydrodynamic force can be decomposed as
_ €
_ €
F
y
t; y; y; y
¼
F
h
ðyÞ þ
F
r
t; y; y
þ
F
d
ðtÞ;
(2)
Fig. 1.
Generic two-body wave energy converter oscillating in heave.
where
F
h
is the hydrostatic restoring force, assumed to be zero
when
y
¼
0,
F
r
is the force exerted by
fluid
on the body as a result of
its oscillation in the absence of incident waves and
F
d
is the vertical
force produced by the incident waves on the
floater fixed
at the
y
¼
0 position.
J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40
33
In the linearized hydrostatic modelling, it is
F
h
¼ À9gSy;
(3)
A
u
j
¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
À Á
2
Du
j
S
z
u
j
;
(7)
(8)
j
¼
2;
.;
n;
(9)
where
9
is the water density,
g
is gravity acceleration and
S
is the
floater’s
waterplane area.
The coefficients of the radiation force are computed in the time
domain as
Du
j
¼ ð1 þ
6randðÞÞ
Du
;
u
j
¼
u
jÀ1
þ
Á
1
À
Du
j
þ
Du
jÀ1
;
2
_ €
F
r
t; y; y
where
À Á
€
_
¼ ÀF
c
t; y
À
A
N
y
ðtÞ;
1
Z
t
ÀN
(4)
À Á
_
F
c
t; y
¼
A
N
1
_
Kðt
À
s
Þ
yð
s
Þd
s
;
(5)
where
u
1
¼
0:1 rad/s,
Du
¼
3:0=n rad/s,
6
¼
0:2 and randðÞ is
a random number generator between 0 and 1. All the spectra
S
z
ð
u
Þ
calculations were performed assuming
n
¼
300 spectrum
components.
For axisymmetric bodies oscillating in heave, the excitation
force coefficient is related to the radiation damping coefficient
by
[21]
and
¼
lim
A
1
ð
u
Þ,
where
A
1
ð
u
Þ
is the frequency dependent
u
/N
added mass coefficient for the
floater.
Since we are assuming linear water wave theory, the resulting
diffraction force is obtained as a superposition of the frequency
components
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
29g
3
Bð
u
Þ
G
ð
u
Þ ¼
:
3
u
(10)
F
d
ðtÞ ¼
with
n
X À Á
j
¼
1
In the current work, real irregular waves are represented as
a superposition of regular waves assuming the Pierson-Moskowitz
[22]
spectrum
G
u
j
A
u
j
cos
u
j
t
þ
2
p
randðÞ
;
À
Á
(6)
H
2
S
z
ð
u
Þ ¼
262:9
5
s
4
exp
u
T
e
!
1054
À
4 4
:
u
T
e
(11)
a
b
Fig. 2.
Diffraction force using
n
¼
300 spectrum components and (a) equally spaced frequencies,
Du
j
¼
Du
, (b) non-equally spaced frequencies, Eq.
(8).
In case (a), the diffraction
force is almost periodic repeating approximately each 660 s.
34
J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40
Numerical results obtained for equally spaced frequencies and
300 spectrum components show an almost periodic diffraction
force pattern approximately each 660 s, see
Fig. 2a.
A large increase
in the probability of getting non-periodic series for longer time
periods was obtained in the current approach by using the same
number of spectrum components with non-equally spaced
frequencies, see
Fig. 2b.
The PTO is modelled in the present work as a linear damper with
coefficient
C
PTO
, together with a spring of stiffness
k
¼
9gS=10
introduced to avoid the vertical drift of the submerged body
À
Á
À
Á
_ _
_
_
F
PTO
y; x
¼
C
PTO
y
À
x
þ
kðy
À
xÞ:
À
Á
À
Á
_
_
_ _
F
brake
t; y; x
¼
C
brake
ðtÞ
y
À
x
:
(12)
The braking force is applied only for latching and is modelled as
(13)
Realistically, in the case of latching, a braking mechanism is
introduced with a non-zero time response. The brake damping
coefficient is defined as a cubic function of time to ensure
a continuous transition between zero and the maximum value
(
C
brake
ðtÞ ¼
2
3
t
t
C
max
3
~
À
2
~
;
C
max
;
0
~
t
~
t>1
1
;
(14)
with
~
¼ ðt À
t
b
Þ=t
max
, where
t
max
is target braking time and
t
À
t
b
t
is the elapsed time since the braking command instant.
The equations of motion for the two-body system in the time
domain are then written as
Fig. 3.
Stages of the threshold unlatching algorithm written in Cþþ. The code is
executed once at the end of each time step.
Ds
.
À
Á
À
Á
_
x
_
€
m
1
þ
A
N
y
þ
C
T
ðtÞ
y
À
À
þ
9gSy
þ
kðy
À
xÞ
¼
F
s
ðtÞ;
1
Á
_
_
x
ðm
2
þ
A
2
Þ
€
þ
C
T
ðtÞ
x
À
y
þ
kðx
À
yÞ
¼
0;
with
(15)
_
_
y
¼
y
0
þ
n
X
i
¼
2
n
X
i
¼
2
ia
i
s
iÀ1
;
(21)
F
s
ðtÞ ¼
F
d
ðtÞ À
F
c
ðtÞ;
C
T
ðtÞ ¼
C
PTO
þ
C
brake
ðtÞ:
(16)
(17)
€
y
¼
iði
À
1Þa
i
s
iÀ2
;
(22)
The system of differential equations is subjected to the initial
_
_
_
_
conditions
yð0Þ
¼
y
0
,
yð0Þ
¼
y
0
,
xð0Þ
¼
x
0
and
xð0Þ
¼
x
0
.
3.2. Numerical solution of the equations of motion
The numerical solution of the motion equations are based on the
polynomial approximation presented in
[23],
adopting a more
generic continuous least-squares method for the computation of
the the polynomials coefficients instead of the the semi-analytical
approach of
[23].
The time step will be denoted by
Ds
. From this point onwards we
will refer to a relative time
s
starting at the beginning of the current
absolute time step,
t
n,
such that
and similar equations for
x.
The use of the relative time allows us to impose the initial
_
_
_
_
conditions
yð0Þ
¼
y
0
,
yð0Þ
¼
y
0
,
xð0Þ
¼
x
0
and
xð0Þ
¼
x
0
directly
in the polynomial approximations
(19)
and
(20).
Additionally, we approximate
F
s
ðtÞ
and
C
T
ðtÞ
also by interpo-
lating polynomials of degree
n
at
n
þ
1 equally spaced points,
f
s
ð
s
m
Þ ¼
F
s
ðt
n
þ
s
m
Þ;
(23)
Table 1
Characteristics of the 14 sea states used in the calculation of the average annual
power absorption.
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
H
s
[m]
1.10
1.18
1.23
1.88
1.96
2.07
2.14
3.06
3.18
3.29
4.75
4.91
6.99
8.17
T
e
[s]
5.49
6.50
7.75
6.33
7.97
9.75
11.58
8.03
9.93
11.80
9.84
12.03
11.69
13.91
4
[%]
7.04
12.35
8.17
11.57
20.66
8.61
0.59
9.41
10.07
2.57
4.72
2.81
1.01
0.39
s
¼
t
À
t
n
:
(18)
We approximate the coordinates of the
floater
and the
submerged body by an
n-th
order polynomial
_
y
¼
y
0
þ
y
0
s
þ
n
X
i
¼
2
a
i
s
i
;
(19)
_
x
¼
x
0
þ
x
0
s
þ
n
X
i
¼
2
b
i
s
i
:
(20)
Consequently
J.C.C. Henriques et al. / Renewable Energy 45 (2012) 31e40
35
a
b
c
d
e
f
Fig. 4.
Dimensionless annual averaged power absorption
P
*
, regarding the wave climate of
Table 1,
as function of threshold unlatching time and PTO damping coefficient for: (a)
L
¼
1, (b)
L
¼
2, (c)
L
¼
5, (d)
L
¼
10 and (e)
L
¼
N.
In case (f), it is
L
¼
5, and a single sea state is used (Pierson-Moskowitz spectrum
H
s
¼
2:8 m and
T
e
¼
8:14 s) with the
same annual energy content as the considered wave climate.
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